1. Field of the Invention
This invention pertains to the general field of process dynamics modeling and control. In particular, it relates to a new and improved method and apparatus for on-line control of continuous crystallizers in order to optimize the operating response to perturbations from steady-state.
2. Description of the Prior Art
Mathematical modeling of the dynamic behavior of process parameters is the basic tool of process control. Once a mathematical relationship between measurable parameters is established either empirically or on the basis of the physics and chemistry of the process, intelligent choices can be made to manipulate certain process parameters in order to control others. Control theory and technology provide the necessary procedure and hardware to accomplish desired goals in an optimal way.
In the case of crystallization and related equipment, maximization of the production of crystals of a desired size distribution is the most important process control objective. Since industrial crystallizers operate at steady state under predetermined target conditions proven experimentally to be optimal for a given process, it is very important to be able to maintain such steady-state conditions and to minimize the effect of any perturbation on the rate of production and size distribution of the crystal product. To that end, much research has been devoted toward finding crystallization models and control procedures that are practical for on-line implementation.
Investigators have shown that the crystal size distribution in crystallizers depends on the kinetics of crystal nucleation and growth, as well as on the constraints and geometry of the equipment. In the case of steady state operation of continuous mixed-suspension type of crystallizers, which are widely used in industry and in the laboratory, it has been found that the population density of each crystal size is represented by the relationship EQU n(L)=n.sup.0 exp(-L/GT),
where n(L) is the number of crystals o linear size L per volume of product slurry in a size range L (population density), G is the crystal linear growth rate, T is the crystallizer's holding time, and n.sup.0 is the population density of embryonic crystals (nuclei) of size vanishingly close to zero. In terms of the cumulative members, N(L), of the crystal population distribution per volume of product slurry, the population density n(L) is defined as the limit of the ratio N(L)/ L as L tends to zero. A complete derivation of the relationship given above appears in Chapter 4 of "Theory of Particulate Processes," Alan D. Randolph et al., Academic Press, New York, Second Edition, 1988.
The exponential form of this steady-state model has been valuable in providing a tool for easy calculation of important but otherwise unmeasurable quantities, such as the nuclei population density n.degree. and the growth rate G. By plotting on semi-logarithmic paper the population density n(L) as a function of particle size L, which in practice can be measured with accuracy for all except the most minute sizes (of the order of units of microns), a straight line results, and n.sup.0 and 1/GT can be derived from its intercept and slope, respectively. Since the holding time for a mixed-product-removal crystallizer is equal to the ratio of the crystallizer's volume and the slurry withdrawal rate, the growth rate G can be calculated from the slope of that line. Similarly, the nucleation rate for the system B.sup.0, defined as the product of the nuclei population density n.sup.0 and the growth rate G, can be calculated from the information so derived. Once these parameters are determined for a given process and the related equipment, control theory provides the necessary algorithms based on the corresponding model to effect process behavior by manipulating predetermined operating parameters that best suit the practical needs and limitations of the crystallizer at hand.
The model given above has been confirmed by much experimental data demonstrating the inverse exponential functionality of n(L) versus L. Therefore, for a given crystallizer operating at steady state with given nuclei population density, growth rate and holding time, it has been shown that the number of crystal particles found in any size interval L decreases exponentially as L increases. In a practical sense, this means that most particles in the crystallizer are smaller than the cut-off size of the product, which is purposely chosen large for quality considerations. Since crystal growth results from mass transport of solute at the surface of existing crystals, product yield (that is, the crystal mass corresponding to the fraction of the crystal population larger than the cut-off size) can be improved by reducing the number of fine particles in the crystallizer, so that the solute is more likely to be transported and contribute to the growth of product-size particles. Thus, a fines removal system is commonly included in continuous crystallizer operation and it is recognized practice to adjust (optimize) the classification and removal of fines in order to enhance crystal growth in the product size-range. When present, an overflow removal circuit serves the same purpose and, in addition, it allows a heavier slurry density to exist in the crystallizer.
Notwithstanding this typical particle size distribution, most of the mass in the product slurry (underflow) is found within crystals larger than the cut-off size because of the cubic relationship between the mass of a particle and its linear dimension. For example, a typical industrial crystallizer producing potassium chloride, operating at steady state with a product cut-off size of 150 to 200 microns, will produce a particle size distribution containing approximately 70 percent of the crystal population below the product cut-off range and 70 percent of the crystal mass above it.
The above described model has been used to calculate crystallization parameters and understand the mechanisms of crystal growth in a variety of applications. For instance, in U.S. Pat. No. 4,025,307 (1977), Randolph et al. use the characteristics of the model to determine the stone-forming crystallization properties of urine from individual patients. From a more precise definition of the kinetics of formation and growth of crystals from urine and the effect of potential inhibitors, a tool is made available for controlling the incidence and proliferation of kidney stones.
In a totally different application, U.S. Pat. No. 4,294,807 to Randolph (1981) teaches the use of the same model in a system for removing flue gas desulfurization solids using a lime or limestone slurry scrubbing solution. Through a better understanding of the mechanisms of particle growth derived from the model, a system is provided to increase the particle size and modify the crystal habit in order to aid the filtration of the process solids and improve the quality of the waste product.
Because of the much greater population density of fine particles mentioned above, it follows that from a practical point of view sufficient information to fit and apply the above described model can be gathered by analyzing samples containing fines only (that is, particles in the size range below the product cut-off size) and neglecting the product population density. In U.S. Pat. No. 4,263,010 (1981), Randolph uses this principle to control the steady-state operation of a continuous crystallizer. By sampling the fines removal circuit only, this patent teaches that an accurate semi-log population density versus size plot can be constructed so long as the data are derived from a sequence of size intervals containing at least three percent of the total crystal mass of fines in the crystallizer. Therefore, the model is derived by on-line measurements of a preconditioned, classified sample of the population distribution with a zone sensing or light scattering particle analyzer. A computerized control apparatus selects a sufficiently large fraction from the resulting population density data and applies regression techniques to fit the model and calculate the nuclei population density and growth rate. This information is in turn used to generate control signals to various manipulated control parameters to maintain process conditions or reduce the effect of perturbations to the steady state. This patent is closely related to the present invention, which is an improvement thereof, and is therefore incorporated herein by reference for the purpose of more fully describing the scope of application of this invention.
It is apparent from the foregoing that the control techniques in the prior art have required sophisticated manipulation of the population density data to fit a specific mathematical model. In addition, the quality of the resulting control depends on how suitable that model is for the specific process being simulated. Therefore, it would be very desirable to develop a model-independent method for crystallizer process control, especially if it comprised a simplification of the computational requirements for the apparatus. The present invention addresses this need and utilizes a novel approach to the use of crystal population information for process control purposes.